Electromagnetic wave propagation and absorption in magnetised plasmas: variational formulations and domain decomposition

We consider a model for the propagation and absorption of electromagnetic waves (in the time-harmonic regime) in a magnetised plasma. We present a rigorous derivation of the model and several boundary conditions modelling wave injection into the plasma. Then we propose several variational formulations, mixed and non-mixed, and prove their well-posedness thanks to a theorem by S\'ebelin et~al. Finally, we propose a non-overlapping domain decomposition framework, show its well-posedness and equivalence with the one-domain formulation. These results appear strongly linked to the spectral properties of the plasma dielectric tensor

Adaptative Newton-like Method for Shape Optimization

14 pagesThe aim of this work is to introduce an adaptive strategy to monitor the rate of convergence of a Newton-like method in numerical shape optimization. Such superlinear iterative algorithm are often computationally intensive and the rate of convergence depends on how accurate the numerical solution of the state equation is. The model concerns a cost function depending on the curvature of $\partial \Omega$, the boundary of the unknown domain, and the solution of an integral equation

Theoretical and Numerical Analysis of a Class of Nonlinear Elliptic Equations

In this paper we show the existence of weak solutions for a nonlinear elliptic equations with arbitrary growth of the non linearity and data measure. A numerical algorithm to compute a numerical approximation of the weak solution is discribed and analysed. In a first step a super-solution is computed using a domain decomposition method. Numerical examples are presented and commented

Theoretical and Numerical Analysis of a Class of Nonlinear Elliptic Equations

19 pagesIn this paper we show the existence of weak solutions for a nonlinear elliptic equations with arbitrary growth of the non linearity and data measure. A numerical algorithm to compute a numerical approximation of the weak solution is described and analyzed. In a first step a super-solution is computed using a domain decomposition method. Numerical examples are presented and commented

Shape optimization for inverse electromagnetic casting problems (version longue)

In this paper we present an algorithm for inverse optimization problems concerning electromagnetic casting of molten metals. We are interested in locating suitable inductors around the molten metal so that the equilibrium shape be as near as possible to a desired target shape. A Simultaneous Analysis and Design (SAND) mathematical programming formulation is stated for the inverse problem. The resulting optimization problem is solved with FAIPA, a feasible directions interior-point algorithm

Local existence and uniqueness of the mild solution to the 1D Vlasov-Poisson system with an initial condition of bounded variation

International audienceWe propose a result of local existence and uniqueness of a mild solution to the one-dimensional Vlasov-Poisson system. We establish the result for an initial condition lying in the space W1,1(R2), then we extend it to initial conditions lying in the space BV (R2), without any assumption of continuity, boundedness or compact support

Inductor shape optimization for electromagnetic casting

International audienceThe design of inductors in electromagnetic shaping of molten metals consists in looking for the position and the shape of a set of electric wires such that the induced electromagnetic field makes a given mass of liquid metal acquire a predefined shape. In this paper we formulate an inverse optimization problem where the position and shape of the inductors are defined by a set of design variables. In a first formulation of the inverse optimization problem we minimize the difference between the target and the equilibrium shapes while in a second approach we minimize the L 2 norm of a fictitious surface pressure that makes the target shape to be in mechanical equilibrium. Geometric constraints that prevent the inductors from penetrating the liquid metal are considered in both formulations. The optimization problems are solved using FAIPA, a line search interior-point algorithm for nonlinear optimization. Some examples are presented to show the effectiveness of the proposed approaches

Local Existence and Uniqueness of a Mild Solution to the One Dimensional Vlasov-Poisson System with an Initial Condition of Bounded Variation

We propose a result of local existence and uniqueness of a mild solution to the one-dimensional Vlasov-Poisson system. We establish the result for an initial condition lying in the Sobolev space of integrable functions with integrable derivatives, then we extend it to initial conditions lying in the space of functions of bounded variation, without any assumption of continuity, boundedness or compact support

Shape optimization for inverse electromagnetic casting problems

International audienceIn this article we present an algorithm for inverse optimization problems concerning electromagnetic casting of molten metals. We are interested in locating suitable inductors around the molten metal so that the equilibrium shape will be as near as possible to a desired target shape. A simultaneous analysis and design mathematical programming formulation is stated for the inverse problem. The resulting optimization problem is solved with FAIPA, a feasible directions interior-point algorithm